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Axiom ax-mulass 7687
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7645. Proofs should normally use mulass 7715 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 7582 . . . 4 class
31, 2wcel 1463 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1463 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1463 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 945 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 7589 . . . . 5 class ·
101, 4, 9co 5740 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 5740 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 5740 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 5740 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1314 . 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
158, 14wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Colors of variables: wff set class
This axiom is referenced by:  mulass  7715
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